[Paper Review] Symmetries of Hamiltonian equations and Lambda-constants of motion

Giampaolo Cicogna|arXiv (Cornell University)|Apr 2, 2010

Nonlinear Waves and Solitons4 references3 citations

TL;DR

This paper introduces the concept of a 'Λ-constant of motion' for Hamiltonian systems exhibiting Λ-symmetry, a generalization of standard Lie point symmetries. It establishes that Λ-symmetries lead to systematic reductions of Hamiltonian equations and ensures the existence of conserved quantities under specific conditions, extending Lagrangian Λ-invariance to the Hamiltonian framework with explicit examples.

ABSTRACT

We consider symmetries and perturbed symmetries of canonical Hamiltonian equations of motion. Specifically we consider the case in which the Hamiltonian equations exhibit a Lambda symmetry under some Lie point vector field. After a brief survey of the relationships between standard symmetries and the existence of first integrals, we recall the definition and the properties of Lambda symmetries. We show that in the presence of a Lambda symmetry for the Hamiltonian equations, one can introduce the notion of "Lambda-constant of motion". The presence of a Lambda symmetry leads also to a nice and useful reduction of the form of the equations. We then consider the case in which the Hamiltonian problem is deduced from a Lambda-invariant Lagrangian. We illustrate how the Lagrangian Lambda-invariance is transferred into the Hamiltonian context and show that the Hamiltonian equations are Lambda-symmetric. We also compare the "partial" (Lagrangian) reduction of the Euler-Lagrange equations with the reduction which can be obtained for the Hamiltonian equations. Several examples illustrate and clarify the various situations.

Motivation & Objective

• To extend the theory of symmetries from standard Lie point symmetries to Λ-symmetries in canonical Hamiltonian systems.
• To define and characterize the notion of a 'Λ-constant of motion' analogous to first integrals under Λ-symmetries.
• To investigate the transfer of Λ-invariance from the Lagrangian to the Hamiltonian formalism.
• To compare the reduction of equations via partial Lagrangian reduction with full Hamiltonian reduction under Λ-symmetries.
• To provide illustrative examples demonstrating the practical utility of Λ-symmetries in simplifying and solving Hamiltonian systems.

Proposed method

• Adapts the standard symmetry condition for Hamiltonian equations to the case of Λ-symmetries using the Lie derivative and total time derivative operators.
• Derives the condition for a quantity to be a Λ-constant of motion by generalizing Noether's theorem to Λ-symmetries, involving the matrix Λ and the symmetry generator.
• Uses the Legendre transformation to connect Λ-invariant Lagrangians to Λ-symmetric Hamiltonian systems, ensuring consistency between the two formalisms.
• Applies the method of invariant coordinates (e.g., w = qp exp(q)) to reduce the number of variables in the system, simplifying the equations of motion.
• Employs the condition D_t S = -∇(ΛΦ) to verify Λ-conservation of the divergence-like quantity S, generalizing the standard divergence condition.
• Compares the partial reduction in the Lagrangian framework (via invariants of X^{(L)}) with the full reduction in the Hamiltonian framework (via invariants of X), showing equivalence in special cases.

Experimental results

Research questions

• RQ1 Can the concept of a first integral be generalized to systems with Λ-symmetries, and what is the corresponding conserved quantity?
• RQ2 How does Λ-invariance of a Lagrangian translate into symmetry properties of the corresponding Hamiltonian equations?
• RQ3 What is the relationship between the partial reduction of Euler-Lagrange equations and the full reduction of Hamiltonian equations under Λ-symmetries?
• RQ4 Under what conditions does a Λ-symmetry lead to a complete reduction of the Hamiltonian system?
• RQ5 Can the quantity S defined by ∇·Φ - {H,τ} be generalized to a Λ-constant of motion, and what is its dynamical behavior?

Key findings

• A Λ-constant of motion can be defined for Hamiltonian systems with Λ-symmetry, satisfying the generalized conservation law D_t S = -∇(ΛΦ), extending the standard Noetherian framework.
• When the Lagrangian is Λ-invariant under a vector field X^{(L)}, the corresponding Hamiltonian system inherits Λ-symmetry under the extended vector field X, ensuring consistency between formalisms.
• In the case where the symmetry generator admits a generating function, a complete reduction of the Hamiltonian system is possible, as demonstrated by the reduction of four variables to three in Example 6.
• The quantity S = ∇·Φ - {H,τ} is shown to be a Λ-constant of motion when the symmetry condition holds, with S = -q in Example 7, and its time derivative satisfies the generalized conservation law.
• The Hamiltonian equations in Example 6 are reduced to a system involving w₁, w₂, G, and z, with explicit dynamics: ẇ₁ = w₁w₃, ẇ₂ = w₃ - w₂, ẋG = -G, ż = z + w₂ - w₃, showing significant simplification.
• The partial Lagrangian reduction yields a particular solution (e.g., θ = 0 or w₂ = w₃ = const), which corresponds to a special case of the full Hamiltonian reduction, confirming consistency between the two approaches.